Master thesis defense by Frederik Holdt-Sørensen
Aspects of gravity and conformal field theories in large dimensions
Four-point correlation functions in a general $D$-dimensional conformal field theory are described by conformal blocks which are evaluated in a radial/angular decomposition, a scalar block suggested large $D$ composition, and a further simplified large $D$ configuration.
In this thesis, we outline the basics of Anti-de Sitter and conformal field theories and motivate the correspondence between the two theories. We introduce a Gegenbauer decomposition of the conformal block in the radial quantization, with a symmetric operator location configuration, and by recursion methods, we derive the first levels in the block's decomposition.
Further, we point out that the transformation of the unit box in cross-ratio coordinates does not cover the unit circle in the radial/angular configuration, and relate this to crossing symmetry.
For large $D$ conformal theories another coordinate set related to the conformal invariant cross-ratios is applied to the Casimir equation and then solved for a negligible mixed term, leaving a simple conformal block expression that is symmetric in $\Delta$ and $1-l$.
We compare this expression with the exact conformal block and the radial/angular conformal block for the case of $D=4$ which resembles the structure remarkably.
The conformal large $D$ block can further be simplified by assuming a linear scaling of $\Delta$ and $l$ with a saddle-point integral and a new coordinate set is applied which using the crossing relation can restrict the external scaling dimension to $\Delta_\phi/D>1$.
This type of block has another behavior compared to the previous three versions and therefore the validity of the linear scaling and the saddle-point integration is questioned.
We show that the coordinate systems of the large $D$ block and the radial/angular block are essentially the same and comment on the different large $D$ limits that are implied for the Casimir equation.
We further make suggestional derivations of the conformal blocks, in some of these large $D$ limits.